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ASITEilETIC  THE  SECOND
class of numbers, so again Dedekind overcame his difficulties by
taking refuge in che infinite.
The concept of an ideal is not hard to grasp, although there
is one twist - the more inclusive class divides the less inclusive, as
will be explained in a moment - which shocks common sense.
However, common sense was made to be shocked; had we
nothing less dentable than shock-proof common sense we
should be a race of mongoloid imbeciles. An ideal must do at
least two things: it must leave common (rational) arithmetic
substantially as it is, and it must force the recalcitrant alge-
braic integers to obey that fundamental law of arithmetic -
unique decomposition into primes - which they defy.
The point about a more inclusive class dividing a less inclu-
sive refers to the following phenomenon (and its generalization,
as stated presently). Consider the fact that 2 divides 4 - arith 
metically, that is, without remainder. Instead of this obvious
fact, which leads nowhere if followed into algebraic number
fields, we replace 2 by the class of all its integer multiples, ... ,
 8,  6,  4,  2,0,2,4,6,8, ... As a matter of convenience
we denote this class by (2). In the same way (4) denotes the
class of all integer multiples of 4. Some of the numbers in (4)
are ... ,  16, 12,  8,  4, 0, 8, 12, 16, ... It is now
obvious that (2) is the more inclusive class; hi fact (2) contains
aU the numbers in (4) and in addition (to mention only two)  6
and 6. The fact that (2) contains (4) is symbolized by writing
(2)|(4). It can be seen quite easily that if m, n are any common
whole numbers then (m) \ (n) when, and only when, m divides n.
This might suggest that the notion of common arithmetical
divisibility be replaced by that of class inclusion as just
described. But this replacement would be futile if it failed to
preserve the characteristic properties of arithmetical divisi-
bility. That it does so preserve them can be seen in detail, but
oae instance must suffice. If m divides n, and n divides J, then
m divides I - for example, 12 divides 24 and 24 divides 72, and
12 does in fact divide 72. Transferred to classes, as above, this
becomes: if (m) \ (n) and () j (Z) then (m) j (Z) or, in English, if
the class (m) contains the class (n), and if the class (n) contains
the class (I), then the class (m) contains the class (I) - which
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